Behrend, Roger E., Di Francesco, Philippe and ZinnJustin, Paul 2013. A doublyrefined enumeration of alternating sign matrices and descending plane partitions. Journal of Combinatorial Theory, Series A 120 (2) , pp. 409432. 10.1016/j.jcta.2012.09.004 

PDF
 Submitted PrePrint Version
Download (329kB)  Preview 
Abstract
It was shown recently by the authors that, for any n, there is equality between the distributions of certain triplets of statistics on nxn alternating sign matrices (ASMs) and descending plane partitions (DPPs) with each part at most n. The statistics for an ASM A are the number of generalized inversions in A, the number of 1's in A and the number of 0's to the left of the 1 in the first row of A, and the respective statistics for a DPP D are the number of nonspecial parts in D, the number of special parts in D and the number of n's in D. Here, the result is generalized to include a fourth statistic for each type of object, where this is the number of 0's to the right of the 1 in the last row of an ASM, and the number of (n1)'s plus the number of rows of length n1 in a DPP. This generalization is proved using the known equality of the threestatistic generating functions, together with relations which express each fourstatistic generating function in terms of its threestatistic counterpart. These relations are obtained by applying the DesnanotJacobi identity to determinantal expressions for the generating functions, where the determinants arise from standard methods involving the sixvertex model with domainwall boundary conditions for ASMs, and nonintersecting lattice paths for DPPs.
Item Type:  Article 

Date Type:  Publication 
Status:  Published 
Schools:  Mathematics 
Subjects:  Q Science > QA Mathematics 
Uncontrolled Keywords:  Alternating sign matrices; descending plane partitions; sixvertex model with domainwall boundary conditions; nonintersecting lattice paths; Desnanot–Jacobi identity 
Publisher:  Elsevier 
ISSN:  00973165 
Last Modified:  04 Jun 2017 03:50 
URI:  http://orca.cf.ac.uk/id/eprint/27748 
Citation Data
Cited 2 times in Google Scholar. View in Google Scholar
Cited 2 times in Scopus. View in Scopus. Powered By Scopus® Data
Actions (repository staff only)
Edit Item 